3: Plane and Spherical Trigonometry
- Page ID
- 6804
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 3.1: Introduction
- It is assumed in this chapter that readers are familiar with the usual elementary formulas encountered in introductory trigonometry. We start the chapter with a brief review of the solution of a plane triangle. While most of this will be familiar to readers, it is suggested that it be not skipped over entirely, because the examples in it contain some cautionary notes concerning hidden pitfalls.
- 3.2: Plane Triangles
- This section is to serve as a brief reminder of how to solve a plane triangle. While there may be a temptation to pass rapidly over this section, it does contain a warning that will become even more pertinent in the section on spherical triangles.
- 3.3: Cylindrical and Spherical Coordinates
- This page presents a detailed overview of cylindrical and spherical coordinates in 3D space, clarifying their connections to rectangular coordinates. It defines key terms like radial, polar, and azimuthal angles, highlighting their uniqueness through calculations. The introduction of direction cosines relates to vector orientation.
- 3.4: Velocity and Acceleration Components
- This page covers two-dimensional polar coordinates and their relationship to three-dimensional spherical coordinates, emphasizing the use of boldface symbols for vectors. It explores the dynamics of a point's motion in polar coordinates, detailing the time-dependent changes in radial and transverse unit vectors, and deriving expressions for velocity and acceleration.
- 3.5: Spherical Triangles
- We are fortunate in that we have four formulas at our disposal for the solution of a spherical triangle, and, as with plane triangles, the art of solving a spherical triangle entails understanding which formula is appropriate under given circumstances. Each formula contains four elements (sides and angles), three of which, in a given problem, are assumed to be known, and the fourth is to be determined.
- 3.6: Rotation of Axes, Two Dimensions
- This page examines the relationship between coordinates on two orthogonal axes separated by an angle \(θ\), deriving equations for converting between them, \(x^\prime = x \cos θ + y \sin θ\) and \(y^\prime = -x \sin θ + y \cos θ\). It presents these equations in matrix form and discusses deriving the inverse relations for \(x\) and \(y\), while highlighting the properties of orthogonal matrices and encouraging multiple approaches to ensure consistency in the results.
- 3.7: Rotation of Axes, Three Dimensions. Eulerian Angles
- This page explores orthogonal axes in three-dimensional space using Eulerian angles to describe rotations. It details mathematical transformations through orthogonal matrices, highlighting properties such as unity of determinant, orthogonality, and relationships with transpose matrices. The reader is prompted to verify these properties and address specific computational tasks, promoting accuracy and clarity in calculations of direction cosines and Eulerian angles.
- 3.8: Trigonometrical Formulas
- A reference a set of commonly-used trigonometric formulas i sprovide. Anyone who is regularly engaged in problems in celestial mechanics or related disciplines will be familiar with most of them.
Thumbnails: If \(C \) is acute, then \(A \) and \(B \) are also acute. Since \(A \le C \), imagine that \(A \) is in standard position in the \(xy\)-coordinate plane and that we rotate the terminal side of \(A \) counterclockwise to the terminal side of the larger angle \(C \). If we pick points \((x_{1},y_{1}) \) and \((x_{2},y_{2}) \) on the terminal sides of \(A \) and \(C \), respectively, so that their distance to the origin is the same number \(r \), then we see from the picture that \(y_{1} \le y_{2} \). Image constructed by Michael Corral (Schoolcraft College).